The Mergelyan-Bishop Theorem

We define the differential operator ∂ ∂z on infinitely differentiable functions (also called smooth or C∞ functions) on some open set in C by ∂ ∂z = 1 2 ( ∂ ∂x + i ∂ ∂y ). A quick calculation shows that ∂ ∂z obeys the product rule. Recall that a function f is holomorphic if and only if ∂ ∂z (f) = 0. A function is biholomorphic, or an analytic isomorphism, if it is holomorphic and has holomorphic inverse. The inverse function theorem of complex analysis tells us that a holomorphic function is biholomorphic in some neighborhood of any point at which its derivative does not vanish. The notation G b X means G is relatively compact in X; that is, the closure of G is compact and contained in X. A Riemann surface is a (connected) one-dimensional complex manifold. An open Riemann surface is a noncompact Riemann surface. We recall a familiar theorem from complex analysis: